Journal
APPLIED MATHEMATICAL MODELLING
Volume 36, Issue 7, Pages 3001-3022Publisher
ELSEVIER SCIENCE INC
DOI: 10.1016/j.apm.2011.09.081
Keywords
Risk measure; Absolute semi-deviation; Fuzzy portfolio optimization; Fractional programming; Pseudo-convexity; Domain decomposition method
Funding
- Natural Science Foundation of Hebei Province [A2011201007]
- Annual Foundation of Social Science Research of Hebei Colleges Universities [SKZD2011302]
- National Natural Science Foundation of China [60974134]
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Since the observed values of security returns in real-world problems are sometimes imprecise or vague, an increasing effort in research is devoted to study the properties of risk measures in fuzzy portfolio optimization problems. In this paper, a new risk measure is suggested to gauge the risk resulted from fuzzy uncertainty. For this purpose, the absolute deviation and absolute semi-deviation are first defined for fuzzy variable by nonlinear fuzzy integrals. To compute effectively the absolute semi-deviations of single fuzzy variable as well as its functions, this paper discusses the methods of computing the absolute semi-deviation by classical Lebesgue-Stieltjes (L-S) integral. After that, several useful absolute deviation and absolute semi-deviation formulas are established for common triangular, trapezoidal and normal fuzzy variables. Applying the absolute semi-deviation as a new risk measure in portfolio optimization, three classes of fuzzy portfolio optimization models are developed by combining the absolute semi-deviation with expected value operator and credibility measure. Based on the analytical representation of absolute semi-deviations, the established fuzzy portfolio selection models can be turned into their equivalent piecewise linear or fractional programming problems. Since the absolute semi-deviation is a piecewise fractional function and pseudo-convex on the feasible subregions of deterministic programming models, we take advantage of the structural characteristics to design a domain decomposition method to separate a deterministic programming problem into three convex subproblems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the effectiveness of the solution method. (C) 2011 Elsevier Inc. All rights reserved.
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